poly/middle_product.hpp

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• Last update: 2023-12-29 16:32:29+09:00
• Include: #include "poly/middle_product.hpp"

Depends on

• poly/ntt.hpp

Required by

• linalg/blackbox/vandermonde.hpp
• poly/composition_f_log_1_minus_x.hpp
• poly/multipoint.hpp
• poly/multivar_convolution_cyclic.hpp
• poly/partial_frac_decomposition_1.hpp
• poly/slice_rational_fps.hpp
• poly/typical_divide_conquer.hpp
• seq/count_grid_path.hpp
• string/wildcard_pattern_matching.hpp

Verified with

• test/library_checker/math/multivariate_convolution_cyclic.test.cpp
• test/library_checker/polynomial/multipoint_evaluation.test.cpp
• test/library_checker/polynomial/multipoint_evaluation_on_geom.test.cpp
• test/library_checker/polynomial/multipoint_evaluation_on_geom_2.test.cpp
• test/library_checker/polynomial/polynomial_interpolation.test.cpp
• test/library_checker/polynomial/polynomial_interpolation_on_geom.test.cpp
• test/mytest/chirp_z.test.cpp
• test/mytest/composition_log_1_minus_x.test.cpp
• test/mytest/partial_frac.test.cpp
• test/mytest/slice_rational_fps.test.cpp
• test/mytest/typical_dc.test.cpp
• test/mytest/vandermonde.test.cpp
• test/yukicoder/1575.test.cpp
• test/yukicoder/1796_1.test.cpp
• test/yukicoder/2166.test.cpp
• test/yukicoder/2231.test.cpp
• test/yukicoder/2243.test.cpp
• test/yukicoder/2626.test.cpp
• test_atcoder/abc241h.test.cpp
• test_atcoder/abc260h.test.cpp
• test_atcoder/abc267g.test.cpp
• test_atcoder/abc272_h.test.cpp
• test_atcoder/abc333f.test.cpp
• test_atcoder/agc013e.test.cpp
• test_atcoder/agc058d2.test.cpp

Code

#pragma once

#include "poly/ntt.hpp"

// n, m 次多項式 (n>=m) a, b → n-m 次多項式 c
// c[i] = sum_j b[j]a[i+j]
template <typename mint>
vc<mint> middle_product(vc<mint>& a, vc<mint>& b) {
  assert(len(a) >= len(b));
  if (b.empty()) return vc<mint>(len(a) - len(b) + 1);
  if (min(len(b), len(a) - len(b) + 1) <= 60) {
    return middle_product_naive(a, b);
  }
  if (!(mint::can_ntt())) {
    return middle_product_garner(a, b);
  } else {
    int n = 1 << __lg(2 * len(a) - 1);
    vc<mint> fa(n), fb(n);
    copy(a.begin(), a.end(), fa.begin());
    copy(b.rbegin(), b.rend(), fb.begin());
    ntt(fa, 0), ntt(fb, 0);
    FOR(i, n) fa[i] *= fb[i];
    ntt(fa, 1);
    fa.resize(len(a));
    fa.erase(fa.begin(), fa.begin() + len(b) - 1);
    return fa;
  }
}

template <typename mint>
vc<mint> middle_product_garner(vc<mint>& a, vc<mint> b) {
  int n = len(a), m = len(b);
  if (!n || !m) return {};
  static const long long nttprimes[] = {754974721, 167772161, 469762049};
  using mint0 = modint<754974721>;
  using mint1 = modint<167772161>;
  using mint2 = modint<469762049>;
  vc<mint0> a0(n), b0(m);
  vc<mint1> a1(n), b1(m);
  vc<mint2> a2(n), b2(m);
  FOR(i, n) a0[i] = a[i].val, a1[i] = a[i].val, a2[i] = a[i].val;
  FOR(i, m) b0[i] = b[i].val, b1[i] = b[i].val, b2[i] = b[i].val;
  auto c0 = middle_product<mint0>(a0, b0);
  auto c1 = middle_product<mint1>(a1, b1);
  auto c2 = middle_product<mint2>(a2, b2);
  const long long m01 = 1LL * nttprimes[0] * nttprimes[1];
  const long long m0_inv_m1 = mint1(nttprimes[0]).inverse().val;
  const long long m01_inv_m2 = mint2(m01).inverse().val;
  const int mod = mint::get_mod();
  auto garner = [&](mint0 x0, mint1 x1, mint2 x2) -> mint {
    int r0 = x0.val, r1 = x1.val, r2 = x2.val;
    int v1 = (m0_inv_m1 * (r1 + nttprimes[1] - r0)) % nttprimes[1];
    auto v2 = (mint2(r2) - r0 - mint2(nttprimes[0]) * v1) * mint2(m01_inv_m2);
    return mint(r0 + 1LL * nttprimes[0] * v1 + m01 % mod * v2.val);
  };
  vc<mint> c(len(c0));
  FOR(i, len(c)) c[i] = garner(c0[i], c1[i], c2[i]);
  return c;
}

template <typename mint>
vc<mint> middle_product_naive(vc<mint>& a, vc<mint>& b) {
  vc<mint> res(len(a) - len(b) + 1);
  FOR(i, len(res)) FOR(j, len(b)) res[i] += b[j] * a[i + j];
  return res;
}

#line 2 "poly/middle_product.hpp"

#line 2 "poly/ntt.hpp"

template <class mint>
void ntt(vector<mint>& a, bool inverse) {
  assert(mint::can_ntt());
  const int rank2 = mint::ntt_info().fi;
  const int mod = mint::get_mod();
  static array<mint, 30> root, iroot;
  static array<mint, 30> rate2, irate2;
  static array<mint, 30> rate3, irate3;

  assert(rank2 != -1 && len(a) <= (1 << max(0, rank2)));

  static bool prepared = 0;
  if (!prepared) {
    prepared = 1;
    root[rank2] = mint::ntt_info().se;
    iroot[rank2] = mint(1) / root[rank2];
    FOR_R(i, rank2) {
      root[i] = root[i + 1] * root[i + 1];
      iroot[i] = iroot[i + 1] * iroot[i + 1];
    }
    mint prod = 1, iprod = 1;
    for (int i = 0; i <= rank2 - 2; i++) {
      rate2[i] = root[i + 2] * prod;
      irate2[i] = iroot[i + 2] * iprod;
      prod *= iroot[i + 2];
      iprod *= root[i + 2];
    }
    prod = 1, iprod = 1;
    for (int i = 0; i <= rank2 - 3; i++) {
      rate3[i] = root[i + 3] * prod;
      irate3[i] = iroot[i + 3] * iprod;
      prod *= iroot[i + 3];
      iprod *= root[i + 3];
    }
  }

  int n = int(a.size());
  int h = topbit(n);
  assert(n == 1 << h);
  if (!inverse) {
    int len = 0;
    while (len < h) {
      if (h - len == 1) {
        int p = 1 << (h - len - 1);
        mint rot = 1;
        FOR(s, 1 << len) {
          int offset = s << (h - len);
          FOR(i, p) {
            auto l = a[i + offset];
            auto r = a[i + offset + p] * rot;
            a[i + offset] = l + r;
            a[i + offset + p] = l - r;
          }
          rot *= rate2[topbit(~s & -~s)];
        }
        len++;
      } else {
        int p = 1 << (h - len - 2);
        mint rot = 1, imag = root[2];
        for (int s = 0; s < (1 << len); s++) {
          mint rot2 = rot * rot;
          mint rot3 = rot2 * rot;
          int offset = s << (h - len);
          for (int i = 0; i < p; i++) {
            u64 mod2 = u64(mod) * mod;
            u64 a0 = a[i + offset].val;
            u64 a1 = u64(a[i + offset + p].val) * rot.val;
            u64 a2 = u64(a[i + offset + 2 * p].val) * rot2.val;
            u64 a3 = u64(a[i + offset + 3 * p].val) * rot3.val;
            u64 a1na3imag = (a1 + mod2 - a3) % mod * imag.val;
            u64 na2 = mod2 - a2;
            a[i + offset] = a0 + a2 + a1 + a3;
            a[i + offset + 1 * p] = a0 + a2 + (2 * mod2 - (a1 + a3));
            a[i + offset + 2 * p] = a0 + na2 + a1na3imag;
            a[i + offset + 3 * p] = a0 + na2 + (mod2 - a1na3imag);
          }
          rot *= rate3[topbit(~s & -~s)];
        }
        len += 2;
      }
    }
  } else {
    mint coef = mint(1) / mint(len(a));
    FOR(i, len(a)) a[i] *= coef;
    int len = h;
    while (len) {
      if (len == 1) {
        int p = 1 << (h - len);
        mint irot = 1;
        FOR(s, 1 << (len - 1)) {
          int offset = s << (h - len + 1);
          FOR(i, p) {
            u64 l = a[i + offset].val;
            u64 r = a[i + offset + p].val;
            a[i + offset] = l + r;
            a[i + offset + p] = (mod + l - r) * irot.val;
          }
          irot *= irate2[topbit(~s & -~s)];
        }
        len--;
      } else {
        int p = 1 << (h - len);
        mint irot = 1, iimag = iroot[2];
        FOR(s, (1 << (len - 2))) {
          mint irot2 = irot * irot;
          mint irot3 = irot2 * irot;
          int offset = s << (h - len + 2);
          for (int i = 0; i < p; i++) {
            u64 a0 = a[i + offset + 0 * p].val;
            u64 a1 = a[i + offset + 1 * p].val;
            u64 a2 = a[i + offset + 2 * p].val;
            u64 a3 = a[i + offset + 3 * p].val;
            u64 x = (mod + a2 - a3) * iimag.val % mod;
            a[i + offset] = a0 + a1 + a2 + a3;
            a[i + offset + 1 * p] = (a0 + mod - a1 + x) * irot.val;
            a[i + offset + 2 * p] = (a0 + a1 + 2 * mod - a2 - a3) * irot2.val;
            a[i + offset + 3 * p] = (a0 + 2 * mod - a1 - x) * irot3.val;
          }
          irot *= irate3[topbit(~s & -~s)];
        }
        len -= 2;
      }
    }
  }
}
#line 4 "poly/middle_product.hpp"

// n, m 次多項式 (n>=m) a, b → n-m 次多項式 c
// c[i] = sum_j b[j]a[i+j]
template <typename mint>
vc<mint> middle_product(vc<mint>& a, vc<mint>& b) {
  assert(len(a) >= len(b));
  if (b.empty()) return vc<mint>(len(a) - len(b) + 1);
  if (min(len(b), len(a) - len(b) + 1) <= 60) {
    return middle_product_naive(a, b);
  }
  if (!(mint::can_ntt())) {
    return middle_product_garner(a, b);
  } else {
    int n = 1 << __lg(2 * len(a) - 1);
    vc<mint> fa(n), fb(n);
    copy(a.begin(), a.end(), fa.begin());
    copy(b.rbegin(), b.rend(), fb.begin());
    ntt(fa, 0), ntt(fb, 0);
    FOR(i, n) fa[i] *= fb[i];
    ntt(fa, 1);
    fa.resize(len(a));
    fa.erase(fa.begin(), fa.begin() + len(b) - 1);
    return fa;
  }
}

template <typename mint>
vc<mint> middle_product_garner(vc<mint>& a, vc<mint> b) {
  int n = len(a), m = len(b);
  if (!n || !m) return {};
  static const long long nttprimes[] = {754974721, 167772161, 469762049};
  using mint0 = modint<754974721>;
  using mint1 = modint<167772161>;
  using mint2 = modint<469762049>;
  vc<mint0> a0(n), b0(m);
  vc<mint1> a1(n), b1(m);
  vc<mint2> a2(n), b2(m);
  FOR(i, n) a0[i] = a[i].val, a1[i] = a[i].val, a2[i] = a[i].val;
  FOR(i, m) b0[i] = b[i].val, b1[i] = b[i].val, b2[i] = b[i].val;
  auto c0 = middle_product<mint0>(a0, b0);
  auto c1 = middle_product<mint1>(a1, b1);
  auto c2 = middle_product<mint2>(a2, b2);
  const long long m01 = 1LL * nttprimes[0] * nttprimes[1];
  const long long m0_inv_m1 = mint1(nttprimes[0]).inverse().val;
  const long long m01_inv_m2 = mint2(m01).inverse().val;
  const int mod = mint::get_mod();
  auto garner = [&](mint0 x0, mint1 x1, mint2 x2) -> mint {
    int r0 = x0.val, r1 = x1.val, r2 = x2.val;
    int v1 = (m0_inv_m1 * (r1 + nttprimes[1] - r0)) % nttprimes[1];
    auto v2 = (mint2(r2) - r0 - mint2(nttprimes[0]) * v1) * mint2(m01_inv_m2);
    return mint(r0 + 1LL * nttprimes[0] * v1 + m01 % mod * v2.val);
  };
  vc<mint> c(len(c0));
  FOR(i, len(c)) c[i] = garner(c0[i], c1[i], c2[i]);
  return c;
}

template <typename mint>
vc<mint> middle_product_naive(vc<mint>& a, vc<mint>& b) {
  vc<mint> res(len(a) - len(b) + 1);
  FOR(i, len(res)) FOR(j, len(b)) res[i] += b[j] * a[i + j];
  return res;
}

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